University of Florida/Egm4313/s12.team8.dupre/R6.1

R6.1

Problem Statement edit

Given  :

(1) Find the (smallest) period of   and  .

(2) Show that these functions also have period  .

(3) Show that the constant   is also a periodic function with period  .

Solution (1) edit

We know that the period of a normal   or   is  . When there are values or variables being multiplied by this   variable, the period becomes   divided by the values or variables. We know that:

  (6.1.1)

Where   is the period.

Using this relation, along with our variable  , we can solve for the period of   and   as follows:

 

  (6.1.2)

Since the period will be smallest at  , plugging into equation (6.1.2) shows that the smallest period of   and   is:

      (6.1.3)

Solution (2) edit

We are also given that:

  (6.1.4)

Using this relation, we can solve for the period again as follows:

 

  (6.1.5)

We know that the period is smallest at   , and plugging this value into (6.1.5) proves that:

      (6.1.6) 

Solution (3) edit

We know that, starting at 0:

  (6.1.7)

Where the period is represented by 0 to 2L. We are also given that:

  (6.1.8)

Rearranging (6.1.8) allows us to solve for L:

  (6.1.9)

Multiplying (6.1.9) by 2 allows us to find the period of  , as follows:

      (6.1.10)

This shows that   is indeed a periodic function with a period of  . This also shows that at any given   value or period throughout the periodic function,   holds its constant value.